There's a nice formula for the Hilbert series of any complete intersection of hypersurfaces $X_1\cap\cdots\cap X_i\subseteq\mathbb{P}^n$ in terms of the degrees of $X_1,\ldots,X_i$. Is there a way to compute the Hilbert series of an irreducible component of $X_1\cap\cdots\cap X_i$ in terms of the Hilbert series of $X_1\cap\cdots\cap X_i$?

More precisely, let $k$ be a field. Suppose $f_1,\ldots,f_i\in R:=k[x_0,\ldots,x_n]$ are homogeneous of degrees $d_1,\ldots,d_i$. Also suppose that $(f_1,\ldots,f_i)$ is a regular sequence in $R$, so that the ideal $I:=(f_1,\ldots,f_i)$ defines a complete intersection $X\subseteq\mathbb{P}^n$ of dimension $n-i$. Since $I$ is homogeneous, we can look at its Hilbert series $HS_I$. Any minimal prime $\mathfrak{p}$ associated to $I$ is also homogeneous, so we can take its Hilbert series $HS_\mathfrak{p}$.

Question: Is there any tractable relationship between $HS_I$ and $HS_\mathfrak{p}$?

I'm actually interested in computing (an upper bound on) $\dim_k\mathfrak{p}_{d}$ in terms of $d_1,\ldots,d_i$ for any $d\leq\min\{ d_1,\ldots,d_i \}$, and Hilbert series seem like the best tool to use. One could try fiddling around with Gröbner bases, but I haven't been able to see how to make this work as generally as I would like.

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    $\begingroup$ What if each $f_i$ is a linear polynomial times a generic polynomial of degree $d_i-1$, and $\mathfrak p$ is the ideal generated by these linear polynomials? This will make $\dim_k \mathfrak p_d$ very large, regardless of the $d_i$. $\endgroup$ – Will Sawin Oct 12 '20 at 15:23
  • $\begingroup$ @WillSawin Thanks, this was a helpful example to think through! I can still get the bound I need in this example, and it's given me some new ideas for the general case. $\endgroup$ – Stephen McKean Oct 12 '20 at 20:24

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